Additive-multiplicative magic squares, 10th-order and above

After additive-multiplicative magic squares of orders 8-9 (also called addition-multiplication magic squares, or even shorter "add-mult"), what about bigger squares?

For example here is my semi-magic add-mult magic square of order 10, the two diagonals being multiplicative magic but unfortunately not additive magic:

I am not fully satisfied with this square because the two diagonals are not additive magic, and because the numbers are big. Who will construct an add-mult fully magic square of order 10? (and if possible with smaller integers!).

Here is another example of my add-mult magic squares. This time the two diagonals are add-mult magic, and the numbers are relatively small:

Table of known add-mult magic squares


"Problem 6.3. For what orders n do addition-multiplication magic squares exist?"
Jószef Dénes (1932-2002) and A. Donald Keedwell, Latin Squares and their Applications (1974), page 489.

Paul Erdös (1913-1996), photo of 1992
"Many unsolved problems are stated, some classical, some due to the authors, and even some proposed by the writer of this foreword."
Paul Erdös, foreword of the above Dénes - Keedwell book (1974) , page 5.

In this Dénes - Keedwell book, the Horner's 8x8 and 9x9 add-mult squares are published pages 215-216 and the above quoted problem 6.3 is posed. I do not have the full answer to this Dénes - Keedwell - Erdös problem, but here is a partial answer with a summary of the best known add-mult magic squares, that you can download with the below Excel file. When their book was published, only orders 8 and 9 were known. We can remark below that orders p and 2p (where p is a prime) are difficult to get.

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